The square root of two cannot exactly be written out on a computer screen in decimal notation – the digits extend forever. You're looking at an approximation accurate to a lot of digits. The approximation itself, however, is rational, as you have demonstrated. If you expanded it out to more digits, however, you'd realize that your quantity has zeros extending out to infinity, while $ \sqrt{2} $ has more digits. Therefore your rational number does not contradict the proof given by the mathematical community that $ \sqrt{2} $ is irrational.
Note: just because a number is irrational doesn't mean it can't be arbitrarily well approximated by rational numbers – every number, rational and irrational can, and that property is known as "the rational numbers are dense in the reals." For instance, for $ \sqrt{2} $:
\begin{align}
\frac{1}{1}, \frac{14}{10}, \frac{141}{100}, \frac{1414}{1000}, \frac{14142}{10000}, ...
\end{align}
get arbitrarily close to $ \sqrt{2} $, but none of them are equal to it.
Related topic: if you're interested in rational numbers which quickly converge to irrational numbers, check out continued fractions. They're really cool. Interesting numbers like $ \sqrt{2} $ often have continued fraction expansions with closed-form coefficients – $ \sqrt{2} $ can be represented by a continued fraction with first coefficient 1 and all following coefficients 2. You can do all sorts of weird things with this, like show that the Golden Ratio (which has a continued fraction expansion of all 1's) is (one of) the irrational number(s) which takes the longest to converge to, or in more plain concepts, the rational numbers used to approximate it require a really large denominator to do a good job.